# return map in a sentence

- A periodic orbit of a flow is said to be hyperbolic if none of the eigenvalues of the Poincar?
*return map*at a point on the orbit has absolute value one. - This is opposed to the lines bx + c . . . also if measured velocity was in fact discretised, wouldn't the
*return map*break up into such clusters? - The attractor was first observed in simulations, then realized physically after Leon Chua invented the Poincar?
*return maps*of the attractor explicitly derived by way of compositions of the eigenvectors of the 3-dimensional state space. - I've shown two examples of the
*return maps*I have gotten for all my Drosophila populations ( more or less in the same form ) to the right, though the acceleration plots and the velocity plots are from different populations. - I was inspired by the discussion here on how to extract nonlinear order from chaos in seemingly random time series such as the logistic map by using something called a " ( Poincare )
*return map*" ( a little different from the Poincare map described here on Wikipedia ). - It's difficult to find
*return map*in a sentence. - Also a reason why I argue against noise is that angular velocity's
*return map*does not break up into discrete lines-- it's a semi-random map with a weak / moderate f ( t + 1 ) =-f ( t ) signal . talk ) 21 : 45, 2 August 2012 ( UTC) - But now that acceleration is close to zero, it will most likely choose acceleration values anywhere from 0 to 500 mm / s ^ 2, so it would prefer one of the two modes ( increasing or maintaining velocity ) but not the decreasing velocity mode, such that velocity will be anywhere from 7 to 43 mm / s at t = 4, but not below that, even though the velocity
*return map*would allow for values between 0 to 7 mm / s otherwise. - At t = 5, it is likely the fly will decelerate ( according to the acceleration
*return map*, because it just accelerated ) proportional to the acceleration chosen-- this locks out the increasing-velocity mode-- but because v can be anywhere from 7 to 43 mm / s ( roughly ) and acceleration anywhere from 0 to 500 mm / s ^ 2 at t = 5, we can't deduce much about t = 6 . ( The acceleration return map cuts off because of how I cropped it but the lines reach + /-500 mm / s ^ 2 .) - At t = 5, it is likely the fly will decelerate ( according to the acceleration return map, because it just accelerated ) proportional to the acceleration chosen-- this locks out the increasing-velocity mode-- but because v can be anywhere from 7 to 43 mm / s ( roughly ) and acceleration anywhere from 0 to 500 mm / s ^ 2 at t = 5, we can't deduce much about t = 6 . ( The acceleration
*return map*cuts off because of how I cropped it but the lines reach + /-500 mm / s ^ 2 .) - I created the
*return maps*by plotting f ( t + 1 ) on the y-axis versus f ( t ) on the x-axis, where f is either velocity or acceleration as appropriate . ( The time difference between time " t " and " t + 1 " is 1 / 15 of a second, i . e . time-series measurements are sampled at 15 Hz . ) It reveals intriguing semidiscrete stochastic decision-making on the part of Drosophila, but how do I even statistically analyse the multiple trendlines, especially so I can detect differences between genetically-different populations? - Of course, if v = 30 mm / s at t = 1, it's v could also be 30 or 60 mm / s at t = 2 ( with less probability ), going into loops that will be broken by drift . Walkthrough : If velocity stays the same ( at 30 mm / s ) at t = 2, acceleration is zero at t = 2, so it will likely accelerate at t = 3, in which case the velocity
*return map*indicates ~ 60 mm / s ^ 2, in which case acceleration will be roughly + 450 mm / s ^ 2 at t = 3. - :: : : : : : I don't think so . . . ? ( I haven't heard of it, at least . . . ) From other papers I've seen where they looked at behavior through
*return maps*, discrete locomotion behaviors ( trot, gallop, etc . ) would correspond to spherical'ish " clusters " on a return map, not lines . i . e . at f ( t ) there would be discrete clusters a, b, c . . . ( corresponding to the different velocity modes ), and at f ( t + 1 ) a, b, c, and the return map would simply imply switching between discrete modes. - :: : : : : : I don't think so . . . ? ( I haven't heard of it, at least . . . ) From other papers I've seen where they looked at behavior through return maps, discrete locomotion behaviors ( trot, gallop, etc . ) would correspond to spherical'ish " clusters " on a
*return map*, not lines . i . e . at f ( t ) there would be discrete clusters a, b, c . . . ( corresponding to the different velocity modes ), and at f ( t + 1 ) a, b, c, and the return map would simply imply switching between discrete modes. - :: : : : : : I don't think so . . . ? ( I haven't heard of it, at least . . . ) From other papers I've seen where they looked at behavior through return maps, discrete locomotion behaviors ( trot, gallop, etc . ) would correspond to spherical'ish " clusters " on a return map, not lines . i . e . at f ( t ) there would be discrete clusters a, b, c . . . ( corresponding to the different velocity modes ), and at f ( t + 1 ) a, b, c, and the
*return map*would simply imply switching between discrete modes. - But this implies ( according to the acceleration
*return map*) that acceleration is zero at t = 5, i . e . at t = 5 v is 30 mm / s ^ 2 with acceleration at t = 6 being anywhere from 0 to 500 mm / s ^ 2, but with a bias to accelerate it at + 450 mm / s ^ 2 towards 60 mm / s at t = 6 for the cycle to begin again . ( Stochastic drift, or chance of being an outlier, allows the fly to escape this cycle . ) But, if v = 60 mm / s at t = 2, then a = ~ + 450 mm / s ^ 2 at t = 2, which means a ~ =-450 mm / s ^ 2 and v ~ = 30 mm / s at t = 3, a ~ = 0 mm / s ^ 2 and v ~ = ~ 30 mm / s at t = 4 and v = 60 mm / s and a = 450 mm / s ^ 2 at t = 5-- another short-term loop which will be broken by drift.